Stronger MIP formulations for the Steiner forest problem. (English) Zbl 1459.90188

Summary: The Steiner forest problem asks for a minimum weight forest that spans a given number of terminal sets. We propose new cut- and flow-based integer linear programming formulations for the problem which yield stronger linear programming bounds than the two previous strongest formulations: The directed cut formulation [A. Balakrishnan et al., Oper. Res. 37, No. 5, 716–740 (1989; Zbl 0681.90083); S. Chopra and M. R. Rao, Math. Program. 64, No. 2 (A), 209–229 (1994; Zbl 0821.90124)] and the advanced flow formulation by T. L. Magnanti and S. Raghavan [Networks 45, No. 2, 61–79 (2005; Zbl 1067.68104)]. We further introduce strengthening constraints and provide an example where the integrality gap of our models is 1.5. In an experimental evaluation, we show that the linear programming bounds of the new formulations are indeed strong on practical instances and that the related branch-and-cut algorithm outperforms algorithms based on the previous formulations.


90C27 Combinatorial optimization
90C57 Polyhedral combinatorics, branch-and-bound, branch-and-cut
90B10 Deterministic network models in operations research


SteinLib; OGDF
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